用数学归纳法证明:1×2+2×5+.+n(3n-1)=n^2(n+1)

证明:当n=1时,左边=1×2=2;右边=1²×(1+1)=2左边=右边,等式成立。假设当n=k时,等式成立,即:1×2+2×5+。+k(3k-1)=k²(k+1)那么,当n=k+1时,1×2+2×5+。+k(3k-1)+(k+1)[3(k+1)-1]=k²(k+1)+(k+1)(3k+2)=(k+1)(k²+3k+2)=(k+1)(k+1)(k+2)=(k+1)²[(k+1)+1]也就是说,当n=k+1时,等式也成立。证毕!